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Let  R  be a commutative ring with identity,be the multiplicative monoid of regular elements in  R ,  t  be the so-called  t -operation on  R  orA Marot ring is a ring whose regular ideals are generated by their regular elements. Marot rings were introduced by J. Marot in 1969 and have been playing a key role in the study of rings with zero divisors. The notion of Marot rings can be extended to  t -Marot rings such that Marot rings are  t -Marot rings. In this paper, we study some ideal-theoretic relationships between a  t -Marot ring  R  and the monoidWe first construct an example of a  t -Marot ring that is not Marot. This also serves as an example of a rank-one DVR of reg-dimensionLet  R  be a  t -Marot ring,  t -(resp.,  t -) be the set of regular prime  t -ideals of  R  (resp., the set of non-empty prime  t -ideals of), andbe the class group of  A  for  A  =  R  orThen, among other things, we prove that the map--given byis bijective;and  R  is a factorial ring if and only ifis a factorial monoid.

The monoid of regular elements in commutative rings with zero divisors